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We consider the Cauchy problems in the whole space for wave equations withhigher derivative terms. We derive sharp growth estimates of the $L^2$-norm ofthe solution itself in the case of the space 1, 2 dimensions. By imposing theweighted $L^1$-initial velocity, we can get the lower and upper bound estimatesof the solution itself. In three or more dimensions, we observe that the$L^2$-growth behavior of the solution never occurs in the ($L^2 \capL^1$)-framework of the initial data.

$L^2$-growth property for wave equations with higher derivative terms